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Fuzzy weighted average: An improved algorithm
Fuzzy Sets and Systems 49 (1992) 307-315 North-Holland
307
Fuzzy weighted average: An improved algorithm T i a n - S h y L i o u a n d M a o - J i u n J. W a n g Department o f Industrial Engineering, National Tsing Hua University, Hsin, Chu, Taiwan, R O C
Received July 1990 Revised April 1991 Abstract: This paper suggests a modification on the fuzzy weighted average (FWA) method developed by Dong and Wong
(1987). A steepest descend/ascend method is proposed to facilitate steps (3) and (4) of the FWA to find the minimum/maximum of f ( x l , x 2 , . . . ,xn, Wl, w 2. . . . . Wn)=~7=lWiXi/~7=l Wi. An example used by Dong and Wong is again employed to illustrate the new method. The same results are obtained but the new method requires much less evaluations and computations. The advantages and disadvantages of the two algorithms are discussed. Also, several relevant theorems are stated and discussed in this paper, and the proofs of these theorems are listed in the Appendix. Keywords: Fuzzy weighted average; multiple criteria decision making.
1. Introduction
In multiple criteria decision making problems, the value of the decision variable is often determined by the problem parameters. When the environment is vague, the rating criteria and their corresponding importance weights are often evaluated in fuzzy numbers. Thus, the value of decision variable is also a fuzzy number which is the fuzzy weighted average of criteria ratings. To generalize the fuzzy weighted average, let At, A 2 . . . . . A,,, and W1, W2. . . . . Wn be the fuzzy numbers defined on the universes X1, X2 . . . . . X~, and Zt, Z2 . . . . . Z. respectively. If f is a function which maps from X~ × X2 x • • • × Xn x Z1 × Z2 x • • • × Zn to the universe Y, then the fuzzy weighted average y is y = f ( x l , x2 . . . . .
xn, wl, w2 . . . . .
(1)
wn) = wlxl + w2x2 +" • • + wnx~ WI -I-- W2 q- . • . -t-- Wn
Let uB be the membership function of the fuzzy image B of A l, f. Then by the extension principle. uB(y) =
max
xi~X~, w ~ Z
{min[uA,(XO . . . . .
UA°(X.). Uw,(WO . . . . .
A2 .....
An,
W~, W2 . . . . .
Wn through
UWo(W.)]}
i-1,2 ..... n y =f (x 1..... x,)
where UA, and Uw, are the membership functions of fuzzy number Ai and W, respectively, i = 1 , 2 . . . . . n. Based on Zadeh's extension principle [5], Dong and Wong [2] proposed an algorithm to compute the fuzzy weighted average (FWA) to the fuzzy number B in y which corresponds to Ai in xi and W~ in wi, i = 1, 2 . . . . . n. The FWA algorithm is summarized as follows: Step 1. Discretize the range of membership [0, 1] into a finite number of values c~, % . . . . .
am. The
degree of accuracy depends on m. Correspondence to: Dr. Mao-Jiun Wang, Department of Industrial Engineering, National Tsing Hua University, Hsin Chu,
Taiwan, 300, ROC. 0165-0114/92/$05.00 ~) 1992--Elsevier Science Publishers B.V. All rights reserved
308
Tian-Shy Liou, Mao-Jiun J. Wang / Fuzzy weighted average
Step 2. For each o:j, find the corresponding intervals for Ai in x~ and W~ in w~, denote the end points of these intervals by [al, bi] and [ci, d~] respectively, i = 1, 2 . . . . . n. These are the supports of the a~j-cuts of A~, A2 . . . . . A , and W~, We . . . . , Wn. If A i or W~ is non-convex, then more than one interval may
result for some of the trj. All these intervals should be treated and the results are the combined union. 22n distinct permutations of the 2n-ary array (Xl, x2 . . . . , x,, wl, w2 . . . . , wn), where x, can be one of the two end points a~ or b~, and w~ can be one of the two end points c~ or d~, i = 1 , 2 . . . . . n. Step 4. Compute Yk = f ( X k ~ , Xk2 . . . . . Xkn, Wkl, Wk2 . . . . . Wkn), where (Xk~, Xkz . . . . . Xkn, Wkl, Wk2, . . . . Wkn) is the k-th permutation of the 2 2n distinct permutations, k = 1, 2 . . . . . 2 2n. Then the desired interval for y is given by [mink Yk, maxk Yk], which is the support of trj-cut of B. Step 5. Repeat steps 2, 3 and 4 for every trj, and the F W A solution to the fuzzy number B is obtained. Step 3. Construct the
Based on the F W A algorithm, for every trj e [0, 1], there is a total of 2n intervals for x~ and w~, i = 1, 2 . . . . . n. Thus 2 2n permutations are involved to compute the interval of y in (1). In other words, for every trj, 2 2n computations for the values Yk, k = 1, 2 . . . . . 2 2n, are required in steps 3 and 4. This becomes very complicated and cumbersome as n increases. In this paper, we propose an improved fuzzy weighted average (IFWA) algorithm to simplify the computational process. Several theorems are applied. The proofs of these theorems are listed in the Appendix.
2. The improved fuzzy weighted average (IFWA) algorithm 2.1. Theoretical b a c k g r o u n d f o r the I F W A algorithm
Consider the weighted average (1) with the fuzzy numbers Ai in xi and W~ in w, for trj e [0, 1]. By step 2 of the F W A algorithm, the corresponding interval for A~ in xi is denoted by [a~, b~], and the corresponding interval for W,. in wi is denoted by [ci, di] for i = 1, 2 . . . . . n. Steps 3 and 4 of the FWA algorithm are to compute the maximum and minimum o f f ( x 1 , x2 . . . . . xn, w~, WE. . . . . W,), where x~ is ai or b~, w~ is ci or di, i = 1, 2 , . . . , n, and thus to obtain the end points of the trj-cut of B. Define fL(Wl, W2. . . . .
Wn) = f ( a l , a2 . . . . .
an, w,, w2. . . .
f v ( w a , WE. . . . .
W,) = f ( b , ,
bn, w,, WE. . . . .
, w~)
and b2,...,
Wn).
Based on the two definitions, the following two theorems hold. The proofs are shown in the Appendix.
Theorem 1. G i v e n x l , x2 . . . . . i --1,
2,
. . . , n,
xn a n d wl, w2 . . . . . a n d wl + w2 + . . . + w , > O, then
fL(Wl, Wz . . . . .
W~) < ~ f ( x l , X2 . . . . .
X,, Wl, W2. . . . .
wn with
ai<~xi<~bi a n d
Wn) <<-ftj(Wl, W2. . . . .
O<~ci<~wi<~di,
for
Wn).
Theorem 2. min/(xl, x2,...
, xn, W l , W2 . . . . .
Wn) = m i n f L ( w a , w2 . . . . .
w,)
and
max ( f ( x l , x2 . . . . .
xn, wl, WE. . . . .
Wn) = m a x f u ( w l , w2 . . . . .
w,).
By Theorem 2, for every tej, the left and right end points of the interval of y can be found by only 2 n evaluations of fL(Wx, WE. . . . . W,) and f u ( w l , WE. . . . . W,) respectively, where wl = ci or di, i = 1, 2 . . . . . n. So we only need 2 ~÷1 evaluations on f ( x ~ , x 2 , . . . , x , , Wl, w2 . . . . , wn) to obtain the interval of y. It is merely 1/2 ~-~ of the total evaluations involved in the F W A algorithm.
T i a n - S h y L i o u , M a o - J i u n J. W a n g
309
/ Fuzzy weighted average
Define f e ( w l , w2 . . . . . Mw,,
w.
wz,...
, w,
[ dk)
=fL(WI .....
W k _ 1, d k ,
I
Wk+ , .....
Wn),
.....
w,,),
,
and similarly f u ( w ~ , wz, . . . , w , I dk) and f u ( w ~ , we . . . . .
w l a l + • • • + W k - l a k _ l + dkak + Wk+lak+~ + • " " + w , a ,
w,, [ d k ) =
f e ( w , , w2 . . . . .
w~ [c,). We have
w~ + • • • + w k - ~ + dk + Wk + l -]-
"
"
"
-t- Wn
W l a 1 -1- • • • + W k _ l a k _ 1 -1- C k a k h- W k + l a k + 1 + • • • q- w n a n -1- ( d k -
=
Ck ) a k
w~ + • • • + Wk-~ + Ck + Wk+~ + " " " + W. + (dk -- Ck)
(2)
That is, the value of fL(Wj, W2, • • • , W. I d~) can be expressed as a fraction of which the numerator and denominator are the sum of numerators and denominators of w l a I -t- • • • q- W k _ l a k _ W 1+"
. .+Wk_
1 q-
Ckak + Wk+~ak+~ + " " •
l+c
kWwk+l+'"
which is the value of fe(w~, w2. . . . .
.+W
+ Wnan
n
wn I ck), and
(dk - Ck)ak dk -- Ck
respectively. When the value of wk changes from ck to dk, the value of fL changes from W, I Ck) to the right side of (2). Before continuing further exploration, a lemma is given. Since the proof is simply an arithmetic operation, it is omitted.
fL(W~, WZ. . . . .
Lemma 1. T h e f o l l o w i n g relations h o l d , p r o v i d e d that a > 0 a n d c > O. (a) I f b /a > d / c , then (b + d ) / ( a + c) < b /a, a n d i f b /a < d / c , then (b + d ) / ( a + c) > b /a. (b) F u r t h e r s u p p o s e a > c a n d b > d. T h e n b / a > d / c w h i c h i m p l i e s (b - d ) / ( a - c) > b / a ,
and
b /a < d / c w h i c h i m p l i e s (b - d ) / ( a - c) < b /a.
In f L ( W l , W2 . . . . . equivalent to (dk
--
dk-
Wn), we have ak = (d~ - c k ) a k / d k -- Ck. If ak
ck)ak< wlal
d- • • • "Jr" W k _ l a k _ 1 "4- C k a k -~ W k + l a k + 1 -t- • • • -~- W n a n
ck
wt + • • • + Wk_l + ck + wk+j + • • • + w ,
By Lemma l(a), we have wla 1 + • • • + Wk_lak_ W 1 +
1+
cgag +
Wk+lak+
1 + • • • + w.a.
" " " Jr- W k _ 1 " ~ C k " ~ W k + 1 "]- " " " " ~ W n
< wla I + • • • + Wk_lak_
1 d- C k a k + W k + l a k +
"q-
+ (d k -
(dk - -
Ck)ak
Ck )
1 + • • • q- w n a n
W l .~_ . . . ~t- W k _ l ..[- C k ..~ W k + l -]_ , o , .~_ W n
that is fL(W~, W2, . . . , W, I dk)fu(w~, w2 . . . . . f u ( w l , w2 . . . . . w , [ dk) > f u ( w l , w2. . . . . w, I ck), and we have the following theorem: Theorem 3. I f ak f u ( w , , w2 . . . . . w n ] c , ) then f u ( w l , w2 . . . . . wn ] d~) > f u ( w l , W e , . . . , Wn [ Ck)"
wn I Ck) then
w~ I Ck), a n d i f
From Theorem 3, by starting from the evaluation of fL(C~, C2 Cn) , we can find the minimum of fL. It will diminish the value of fL by replacing Ck with dk, then ak is less than the present value of fLSimilarly, in finding the maximum o f f u , we can start with evaluating f u ( c t , c2 . . . . . c,); if bk is greater .
.
.
.
.
Tian-Shy Liou, Mao-Jiun J. Wang / Fuzzy weighted average
310
than the present value of fu, then replace Ck with dk and this will increase the fu value. It is impossible to replace dk with Ck again when Ck has been replaced by dk. This is summarized in Theorem 4. Theorem 4. In the processes of finding the minimum of fL and the maximum of fv, Ck can never replace
dk again once it has been replaced by dk. Now, the last question needed to be answered in the IFWA algorithm is which candidate should be selected in replacing Ck by dk when more than one candidate is found. Let RL and Ru be the sets of the index i for which w~= di in fL(Wl, W2. . . . , Wn) and fu(w~, w2. . . . . w~) respectively. That is, wi = di for i e RL (Rtj) and wi = ci for i ~ RL (Rv). The selection rules are described in Theorem 5. T h e o r e m 5. In the process of finding the minimum of fL, suppose that L =fL(W~, W2. . . . .
Wn), where wi = di for i ~ RL and wi = ci for i ~ RL, is the present value of fL. Let I = {i I ai < L, i ~ RL} and fL( wl' W2 . . . . .
Wn I din)
-- m i n f L ( w l ,
i~1
w2. . . . .
wn
I di),
then dm is to be selected to replace C m . Similarly, in the process of finding the maximum of fv, suppose that U =fu(wl, wz, • . •, wn), where wi = di for i ~ R u and wi = ci for i ~ Ru, is the present value of fu. Let J = {j I bi > U, i ~ R v } and f u ( wl' W2 . . . . .
Wn I din) = m a x f u ( w l , w2 . . . . jeJ
, wn l d J) ,
then d,, is to be selected to replace c,,. Theorem 5 provides not only the selection rules when more than one index is involved in I or J, but it also provides the stopping rule for the IFWA algorithm when both I and J are empty. 2.2. IFWA algorithm The IFWA algorithm is developed by modifying steps 3 and 4 of the FWA algorithm. The algorithm is described in the following:
Step a. Evaluate L = f L ( C l , C2, . . • , C n ) and U = f u ( C l , C 2 . . . . . C n ) . Let I = {i I ai < L, i = 1, 2 . . . . . n}, J={j[bj>U,j =l,2,...,n} andRL=RU=O. Step b. If I = 0 then L is the minimum of fL and stop step b. Else w, = di for i ~ RL and wi = c~ for i ~t RL. bl. Evaluate li =fL(Wl, W2. . . . . W, t d~) for i ~ I. b2. L =lm = min/El li, put m into RL and take m away from I; put T = {i [ ai > L, i ~ I}. b3. Take i away from I, where i belongs to T. If 14: 0, then repeat step b, else stop step b and L is the minimum of fL. Step c. If J = 0 then U is the maximum of fu and stop step c. Else w~= d i for i ~ Ru and w/= c~ for i¢Ru. cl. Evaluate uj=ftj(wl, w2 . . . . . wn ]d r) for j eJ. c2. U =Um = maxj~j Uj, put m into Ru and take m away from J; put T = {j I b~ < U, j e J}. c3. Take j away from J, where j belongs to T. If J #=0, then repeat step c, else stop step c and U is the maximum of fv. Step d. [L, U] is the interval of a~i-cut of y. Go to step 5 of the FWA algorithm.
Tian-Shy Liou, Mao-Jiun J. Wang / Fuzzy weighted average
311
3. Example 3.1. Three-term weighted average Consider the weighted average
y = f ( x l , x2, x3, Wl, w2, w 3 ) -
WlX 1 + W2X 2 + W3X 3
wl + w2 + w3
where the fuzzy number A~ in x~, A2 in x2, A3 in x3, WI in w~, W2 in w2, and W3 in w3 are defined as {21,
UAI(X1)=
X1' ~ x 3 - 4,
ua3(x3) = [ 6 - x3 '
0<~Xl< 1, 1 ~
{x2-2, U'42(X2)=I.4--X2 '
4~
Wl/0.3, Uw,(Wl) = ( 0 . 9 - wO/0.6,
0 ~< wl < 0.3, 0.3 ~< wl ~< 0.9,
= ~ ( w 3 - 0.6)/0.2,
0.6~< w3 < 0.8, 0.8 ~< w3 ~< 1.
{(w2-0.4)/0.3, Uw2(Wz)={(1-w2)/0.3,
0.4 ~< w2 < 0.7, 0.7 ~< wz<~ 1,
2~
Uw3(W3) [(1_w3)/0.2,
In the F W A algorthm, it needs 22×3 = 64 evaluations on f for each a~j e [0, 1] to get the c~j-cut of y. For the new I F W A algorithm, with r e = 0 . 5 , the procedure and results are illustrated in the following section. The procedure is the same for all other tr/s.
3.2. Numerical result For tr = 0.5, the intervals of x~, x2, x3, w~, w2 and w3 are [a~ = 0.5, bl = 1.5], [a2 = 2.5, b2 = 3.5], [a 3 = 4.5, b3 = 5.5], [Cl = 0.15, d l = 0.6], [c2 = 0.55, d2 = 0.85], and [c3 = 0.7, d3 = 0.9] respectively. fL(Wl, W2, W3)=
0.5wl + 2.5w2 + 4.5w 3 W 1 + W2 +
and
fu(w~, w2, w3)=
1.5W1+3.5~+5.5%
W3
wl + WE+ W3
The computational procedure is as follows:
Step a: L =ft.(Cl, c2, c3) = fL(0.15, 0.55, 0 . 7 ) =
U = f u ( c l , c2, c3) = fu(0.15, 0.55, 0.7) 1={1,2},
J={3}
and
0.15X0.5+0.55X2.5+0.7X4.5 0.15+0.55+0.7
m
4.6 1.4'
0.15 x 1.5 + 0.55 x 3.5 + 0.7 x 5.5
6
0.15 + 0.55 + 0.7
1.4'
RL=Ru=0.
Step b: I = {1, 2} 4: 0, L = 4.6/1.4, Re = 0, so wl = cl, w2 = c2, w3 = c3. Step b l : l, = M 0 . 1 5 , 0.55, 0.7 I d , ) = fL(0.6, 0.55, 0.7)
4.6+(dl-cl)al 1.4 + (d~ - c1)
4.6+(0.6-0.15)
x0.5
1.4+(0.6-0.15)
4.825 1.85 '
12=•(0.15, 0.55, 0.7 1dz ) =fL(0.15, 0.85, 0.7) 4.6+(de-cz)a2_4.6+(0.85-0.55)×2.5 1.4 + (de - c2) 1.4 + (0.85 - 0.55)
5.35 1.7
Step b2: L = min{4.825/1.85, 5.35/1.7} = 4.825/1.85 = 11, i.e. m = 1, so R L = { 1 } , 1 = { 2 } . T=0.
Step b3: I = {2}, repeat step b.
And
Tian-Shy Liou, Mao-Jiun J. Wang / Fuzzy weighted average
312
Step b: I = {2} 4:0, L = 4.825/1.85, RL = {1}, so Wl = d l , w2 = c2, W3:C3. Step b l : 12 = fL(0.6, 0.55, 0.7 I d2) = fL(0.6, 0.85, 0.7)
Step Step Step Step
4.825 + ( d 2 - c2)a2
4.865 + (0.85 - 0.55) x 2.5
5.575
1.85 + (d2 - c2)
1.85 + (0.85 - 0.55)
2.15
b2: L = min{5.575/2.15} = 5.575/2.15 = 12, i.e. m = 2, so RL = {1, 2}, I = 0. And T = 0. b3: I = O, stop step b and L = 5.575/2.15 = 2.59 is the minimum offL. c: J = {3} 4:0, U = 6/1.4, R u = 0 , so W I = C 1 , W2~--C2, W 3 : C 3. c1:
u3 =fu(0.15, 0.55, 0.7 [ d 3 ) = f u ( 0 . 1 5 , 0.55, 0.9)
6+(d3-c3)b 3 6+(0.9-0.7) 1.4 + (d 3 - c3)
x5.5
1.4 + (0.9 - 0.7)
7.1 1.6"
Step c2: U = max {7.1/1.6} = 7.1/1.6 = u3, i.e. m = 3, so Ru = {3}, J = 0. And T = 0. Step c3: J = 0, stop step c and U = 7.1/1.6 = 4.44 is the maximum o f f u . Step d: [2.59, 4.44] is the interval of the 0.5-cut of y. Go to step 5 of the F W A algorithm. The procedures are repeated for tr = 0 and 0~= 1, and the results obtained are [1.68, 5.43] and [3.56, 3.56] respectively. These results are exactly the same as that of the F W A algorithm (see Dong and Wong [2]).
4. Discussion and conclusion
4. I. Advantage of the I F W A algorithm in calculation In the I F W A algorithm, the worst case is that all the left end-points ai are less than the initial L and all the right end-points bj are greater than the initial U, and no i in T (i.e. T = 0) is away from I in step b3, and no j in T (i.e. T = 0) is away from J in step c3, when steps b and c are repeated. In the worst case, the number of evaluations for fL and fv is 1 + ½n(n + 1) respectively. The total number if n(n + 1) + 2 which is much less than the number required for the F W A algorithm (22n). The former is polynomial time but the latter is exponential time. In evaluating the number of arithmetic operations involved in f, the F W A algorithm needs n multiplications, 2(n - 1) additions, and 1 division. In the I F W A algorithm, no matter how large the n is, there are only 1 multiplication, 1 subtraction, 2 additions, and 1 division. In here, we preserve the numerator and denominator of L, add them to (di-ci)ai and d i - c ~ in step bl to evaluate fL(Wl, W2. . . . . Wn I d~), and preserve the numerator and denominator of U, add them to (di - c~)a~ and di - ci in step cl to evaluate fu(wl, w2. . . . . wn I di) (see (2) and Section 3.2). As can be seen, the number of arithmetic operations has been reduced greatly.
4.2. Advantage of the IFWA algorithms in comparison For the number of comparisons, the I F W A algorithm needs at most n(n + 1) comparisons to find I and J in step a and to find T in steps b2 and c2. Also in steps b2 and c2, the maximum number of comparisons needed to find L and U is n - k (see Horowitz and Sahni [3], the straightforward maximum and minimum method) in the k-th cycle respectively, which has a total of n ( n - 1) comparisons. Overall, the algorithm requires at most 2n 2 comparisons to find the interval of the ag-cut of y. In the F W A algorithm, the maximum and minimum of y~, Y2, - . . , Yz2- are to be found. In analyzing the complexity of an algorithm, the number of element comparisons is always the main concern (see Borodin and Munro [1] and Horowitz and Sahni [3]). In finding the maximum and minimum value,
Tian-Shy Liou, Mao-Jiun J. Wang / Fuzzy weighted average
313
using the divide-and-conquer strategy, when the n u m b e r of elements m is a power of 2, i.e. m = 2 k for some positive integer k, the n u m b e r of comparisons is 3m - 2 . A n d it has been proven that no algorithm involves less than 3m - 2 comparisons (see Horowitz and Sahni [3], and Pohl [4]). In this case, m = 22". So the n u m b e r of element comparisons in the F W A algorithms is no less than 3 ( 2 2 " ) - 2 , that is far greater than 2n 2 comparisons involved in the I F W A algorithm. In summary, the I F W A algorithm which only modifies steps 3 and 4 of the F W A algorithm has been proven to be a more efficient algorithm in terms of the n u m b e r of calculations and comparisons required than that of the F W A algorithm.
Appendix Proof of Theorem 1. Since ai <<-x i <~ b i and w i > O, for i = 1, 2 , . . . , n, we have w i a i ~ w i x i <~ w~b~, for i = 1, 2 . . . . . n. T a k e the summation on the inequality, then divide by wl + w2 + • • • + wn and the proof is completed. Proof of Theorem 2. Since m i n f ( x l , x2, • • • , xn,
wl, w2, • • • , wn) <~f(xl,
x2 ....
, x,,
wl, w2, . • • , w~)
for all , Xn, Wl, WZ . . . .
( X l , X2 . . . .
e{(Xl,X2
.....
x.,wl,
, Wn) w2 . . . . .
we have m i n f ( x l , x 2 . . . . . x,,, wl, i = 1, 2 . . . . . n. This implies that
Wn) l a i < ~ x i < ~ b i ,
w2 . . . . .
m i n f ( x l , x2, . . • , x., wl, w2 . . . . . that is minf(x~,
x2 .....
minfL(w~, w2 . . . . .
w,,) <~f(al,
a2 . . . . .
w , , ) <<-minfL(Wl,
w.) ~< m i n f ( x l , x2 . . . . .
,n},
a., w~, wz .....
w,,)
a., wl, w2 . . . . .
w.),
w.) ~< m i n f ( a l , a2 . . . . .
x,,, wl, wz .....
i=1,2 ....
ci<~wi<~di,
w2 . . . . .
x., w~, w 2 , . . . ,
for all
ci <~ w~ <~ d i ,
w.). Following T h e o r e m 1, we have w.),
and we obtain that m i n f ( x l , x2 . . . . . x., wl, w 2 , . . . , w.) = minfL(wl, w2 . . . . . w.). Similarly, m a x f ( x ~ , xz . . . . . x., w~, w2 . . . . . w.) = m a x f v ( w l , w2 . . . . . w.) can be obtained. Thus T h e o r e m 2 is proven.
Proof of Theorem 4. Without losing generality, assume that R L = { 1 , 2 , . . . , k} and the value of
fL
fc(dl .....
is
fL(dl ..... dk-1,
d k , Ck+l . . . . .
Ck . . . . .
C,,),
d~al + • • • + dk-lak-~
C,)
when
cg
has
been
replaced
by
d~.
Because
ak<
SO
+ ckak + " " " +Cnan a k > O,
d~ + " • • + d k - ~ + C~ + • • • + Cn
which implies that dl(a
1-
ak) +...
4- d k _ l ( a k _ 1 -- a k ) q- Ck+l(ak+ 1 -- a k ) + " • • + Cn(an -- a k ) > O, d~ + .
• • +dk-~
+ck
+" • • +c,
and thus d l ( a l - ak) + ' "
Let RL = {1, 2 , . . .
+ dk_l(a~_~ - ak) + Ck+~(ak+~ -- a~) + . . .
+ c . ( a . -- a~) > 0.
, p } , where p ~
fL(dl .....
(A1) d p , Cp+l . . . . .
cn).
Tian-Shy Liou, Mao-Jiun J. Wang / Fuzzy weighted average
314
Surely, p/> k, and fL(dl . . . . . dl(al
d i a l + " • • + dkak + " " " + dpap + cp+lap+l + • • • + c . a ~
tip, Cp+l, . . . , % ) - - a k = ak) + ' ' "
-
+ dk-l(ak-1
dl + " ' " + d p +Cp+l + ' " • + c.
--ak
-- a k ) -I- d k + l ( a k + 1 -- a k ) + . . .
=
+ d p ( a v - ak) + Cp+l(ap+l - ak) + ' ' '
+ cn(an -- ak)
d l + " • " + dp + Cv+l+" • • +c,, >I d l ( a , - ak) + ' ' "
+ dk-l(ak-1
-- ak) + Ct,+l(ak+l -- ak) + ' ' "
+ c,,(a,, - ak) > 0
d l + " " • + dp + cp+l + " " " + c,,
(see (A1), and d i >! ci, i = k + 1 . . . . . is less than the present value of fL. f
(d, . . . .
,
d k - a , ck, dk+~ . . . . .
dial + • • • + dk-lak-i
p ) . It is proven that ak
dp, cp+l . . . . .
c,,), that is ak
c.)
dp, Cp+l . . . . .
+ Ckak + dk+~ak+l + " " " + dpap + Cp+lap+ 1 + • • • + Cna n
d l + " • " + d k - I + ck + d k + l + " " " + dp + Cp+ 1 "q- " " " -~- Cn
= dial + • • • + dpa. + c.+la.+l dl + " " + df, + Ce+l + >
d l a 1 + . • • + d p a , + Cp+lap+ 1 dl +...+
=fL(dl,
• • • ,
+ " • • + c . a ~ - (dk - c k ) a k •
•
•
+ C. -- (dk -- Ck)
+" • • + %a.
(from Lemma l(b))
dp q-Cp+l + . • . + C n
dk-1, dk, dk+l,..
•,
dp, Cp+l . . . . .
Cn).
This proves that the replacement of dk by Ck would increase the present value of rE- Therefore, in finding the minimum of rE, Ck can never replace d k again once it has been replaced by d k. Similarly, the same reason is true for finding the maximum of fu. The proof of Theorem 4 is completed•
Proof of Theorem 5. To simplify the problem, we suppose that only as and aj are less than the present value of fL(W~, W2 . . . . . W , ) , where wk = d k for k e R~ and wk =Ck for k ~RL. It is obvious that i,j~RL. Then di may replace ci to get fc(Wl, W Z , . . . , w , l d i ) or dj may replace cj to get fL(Wl, WE,...,
W, ] dj).
Assume that f L ( W , , WE. . . . .
w , l di)
(A2)
w~ l di).
Case 1: If we select dj to replace cj, and suppose ai
W~ ] dj), then d~ replaces ci to get
f c ( W l , W2, . . . , Wn I dj, di), and fL(Wl,
WE,...,
Wn I dy, di) ~ f L ( W , , W2, . . . , Wn I d j ) .
Two consequences may occur: (1) f L ( W , , W2, . • • , W , ] dj, di) <~fL(W~, W2, • • • , W, d~); this is what we want• (2) f~(Wl, WE, • • •, W~ I dj, d~) >fL(w~, W2. . . . . Wn d~); this is worse than replacing c~ by di only and it is not a desirable outcome. Case 2: If we select d; to replace c~, the result can be one of two outcomes: (1) aj ~>fL(Wl, W2. . . . . W, [ di); by Theorem 3, dj does not replace cj. (2) aj < fL(W~, W2, , W, ] di); in this c a s e , dj r e p l a c e s cj to g e t f L ( w , , w2 . . . . , wn ] d~, dj) a n d f L ( W I ' W2 . . . . .
Wn I d i ' d j ) < - f L ( W , , W2 . . . . .
By (A2), we have f L ( W l , W2 . . . . .
w , I di)•
W~ ] d~, dj) <~fL(W~, WE . . . . .
W~ [ dj), which is just what we want.
Tian-Shy Liou, Mao-Jiun J. Wang / Fuzzy weighted average
So, in finding fL(Wl, W2. . . . . W. M w ~ , w2 . . . . w. This relationship proven.
the minimum of dj). Similarly, in d 3
315
fL, di is to replace ci when f L ( W l , W2 . . . . , W. 14)< finding the maximum of fu, dj is to replace c~ when w. l dj). the case of more than two candidates, and Theorem 5 is thus
References [1] A. Borodin and I. Munro, The Computational Complexity of Algebraic and Numeric Problems (American Elsevier, New York, 1975). [21 W.M. Dong and F.S. Wong, Fuzzy weighted averages and implementation of the extension principle, Fuzzy Sets and Systems 21 (1987) 183-199. [3] E. Horowitz and S. Sahni, Fundmentals of Computer Algorithms (Computer Science Press, Rockville, MD, 1978) 108-112. [4] I. Pohl, A sorting problem and its complexity, Comm. Assoc. Comput. Mach. 15 (1972) 462-463. [5] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning, Inform. Sci. 8 0975) 199-249.
307
Fuzzy weighted average: An improved algorithm T i a n - S h y L i o u a n d M a o - J i u n J. W a n g Department o f Industrial Engineering, National Tsing Hua University, Hsin, Chu, Taiwan, R O C
Received July 1990 Revised April 1991 Abstract: This paper suggests a modification on the fuzzy weighted average (FWA) method developed by Dong and Wong
(1987). A steepest descend/ascend method is proposed to facilitate steps (3) and (4) of the FWA to find the minimum/maximum of f ( x l , x 2 , . . . ,xn, Wl, w 2. . . . . Wn)=~7=lWiXi/~7=l Wi. An example used by Dong and Wong is again employed to illustrate the new method. The same results are obtained but the new method requires much less evaluations and computations. The advantages and disadvantages of the two algorithms are discussed. Also, several relevant theorems are stated and discussed in this paper, and the proofs of these theorems are listed in the Appendix. Keywords: Fuzzy weighted average; multiple criteria decision making.
1. Introduction
In multiple criteria decision making problems, the value of the decision variable is often determined by the problem parameters. When the environment is vague, the rating criteria and their corresponding importance weights are often evaluated in fuzzy numbers. Thus, the value of decision variable is also a fuzzy number which is the fuzzy weighted average of criteria ratings. To generalize the fuzzy weighted average, let At, A 2 . . . . . A,,, and W1, W2. . . . . Wn be the fuzzy numbers defined on the universes X1, X2 . . . . . X~, and Zt, Z2 . . . . . Z. respectively. If f is a function which maps from X~ × X2 x • • • × Xn x Z1 × Z2 x • • • × Zn to the universe Y, then the fuzzy weighted average y is y = f ( x l , x2 . . . . .
xn, wl, w2 . . . . .
(1)
wn) = wlxl + w2x2 +" • • + wnx~ WI -I-- W2 q- . • . -t-- Wn
Let uB be the membership function of the fuzzy image B of A l, f. Then by the extension principle. uB(y) =
max
xi~X~, w ~ Z
{min[uA,(XO . . . . .
UA°(X.). Uw,(WO . . . . .
A2 .....
An,
W~, W2 . . . . .
Wn through
UWo(W.)]}
i-1,2 ..... n y =f (x 1..... x,)
where UA, and Uw, are the membership functions of fuzzy number Ai and W, respectively, i = 1 , 2 . . . . . n. Based on Zadeh's extension principle [5], Dong and Wong [2] proposed an algorithm to compute the fuzzy weighted average (FWA) to the fuzzy number B in y which corresponds to Ai in xi and W~ in wi, i = 1, 2 . . . . . n. The FWA algorithm is summarized as follows: Step 1. Discretize the range of membership [0, 1] into a finite number of values c~, % . . . . .
am. The
degree of accuracy depends on m. Correspondence to: Dr. Mao-Jiun Wang, Department of Industrial Engineering, National Tsing Hua University, Hsin Chu,
Taiwan, 300, ROC. 0165-0114/92/$05.00 ~) 1992--Elsevier Science Publishers B.V. All rights reserved
308
Tian-Shy Liou, Mao-Jiun J. Wang / Fuzzy weighted average
Step 2. For each o:j, find the corresponding intervals for Ai in x~ and W~ in w~, denote the end points of these intervals by [al, bi] and [ci, d~] respectively, i = 1, 2 . . . . . n. These are the supports of the a~j-cuts of A~, A2 . . . . . A , and W~, We . . . . , Wn. If A i or W~ is non-convex, then more than one interval may
result for some of the trj. All these intervals should be treated and the results are the combined union. 22n distinct permutations of the 2n-ary array (Xl, x2 . . . . , x,, wl, w2 . . . . , wn), where x, can be one of the two end points a~ or b~, and w~ can be one of the two end points c~ or d~, i = 1 , 2 . . . . . n. Step 4. Compute Yk = f ( X k ~ , Xk2 . . . . . Xkn, Wkl, Wk2 . . . . . Wkn), where (Xk~, Xkz . . . . . Xkn, Wkl, Wk2, . . . . Wkn) is the k-th permutation of the 2 2n distinct permutations, k = 1, 2 . . . . . 2 2n. Then the desired interval for y is given by [mink Yk, maxk Yk], which is the support of trj-cut of B. Step 5. Repeat steps 2, 3 and 4 for every trj, and the F W A solution to the fuzzy number B is obtained. Step 3. Construct the
Based on the F W A algorithm, for every trj e [0, 1], there is a total of 2n intervals for x~ and w~, i = 1, 2 . . . . . n. Thus 2 2n permutations are involved to compute the interval of y in (1). In other words, for every trj, 2 2n computations for the values Yk, k = 1, 2 . . . . . 2 2n, are required in steps 3 and 4. This becomes very complicated and cumbersome as n increases. In this paper, we propose an improved fuzzy weighted average (IFWA) algorithm to simplify the computational process. Several theorems are applied. The proofs of these theorems are listed in the Appendix.
2. The improved fuzzy weighted average (IFWA) algorithm 2.1. Theoretical b a c k g r o u n d f o r the I F W A algorithm
Consider the weighted average (1) with the fuzzy numbers Ai in xi and W~ in w, for trj e [0, 1]. By step 2 of the F W A algorithm, the corresponding interval for A~ in xi is denoted by [a~, b~], and the corresponding interval for W,. in wi is denoted by [ci, di] for i = 1, 2 . . . . . n. Steps 3 and 4 of the FWA algorithm are to compute the maximum and minimum o f f ( x 1 , x2 . . . . . xn, w~, WE. . . . . W,), where x~ is ai or b~, w~ is ci or di, i = 1, 2 , . . . , n, and thus to obtain the end points of the trj-cut of B. Define fL(Wl, W2. . . . .
Wn) = f ( a l , a2 . . . . .
an, w,, w2. . . .
f v ( w a , WE. . . . .
W,) = f ( b , ,
bn, w,, WE. . . . .
, w~)
and b2,...,
Wn).
Based on the two definitions, the following two theorems hold. The proofs are shown in the Appendix.
Theorem 1. G i v e n x l , x2 . . . . . i --1,
2,
. . . , n,
xn a n d wl, w2 . . . . . a n d wl + w2 + . . . + w , > O, then
fL(Wl, Wz . . . . .
W~) < ~ f ( x l , X2 . . . . .
X,, Wl, W2. . . . .
wn with
ai<~xi<~bi a n d
Wn) <<-ftj(Wl, W2. . . . .
O<~ci<~wi<~di,
for
Wn).
Theorem 2. min/(xl, x2,...
, xn, W l , W2 . . . . .
Wn) = m i n f L ( w a , w2 . . . . .
w,)
and
max ( f ( x l , x2 . . . . .
xn, wl, WE. . . . .
Wn) = m a x f u ( w l , w2 . . . . .
w,).
By Theorem 2, for every tej, the left and right end points of the interval of y can be found by only 2 n evaluations of fL(Wx, WE. . . . . W,) and f u ( w l , WE. . . . . W,) respectively, where wl = ci or di, i = 1, 2 . . . . . n. So we only need 2 ~÷1 evaluations on f ( x ~ , x 2 , . . . , x , , Wl, w2 . . . . , wn) to obtain the interval of y. It is merely 1/2 ~-~ of the total evaluations involved in the F W A algorithm.
T i a n - S h y L i o u , M a o - J i u n J. W a n g
309
/ Fuzzy weighted average
Define f e ( w l , w2 . . . . . Mw,,
w.
wz,...
, w,
[ dk)
=fL(WI .....
W k _ 1, d k ,
I
Wk+ , .....
Wn),
.....
w,,),
,
and similarly f u ( w ~ , wz, . . . , w , I dk) and f u ( w ~ , we . . . . .
w l a l + • • • + W k - l a k _ l + dkak + Wk+lak+~ + • " " + w , a ,
w,, [ d k ) =
f e ( w , , w2 . . . . .
w~ [c,). We have
w~ + • • • + w k - ~ + dk + Wk + l -]-
"
"
"
-t- Wn
W l a 1 -1- • • • + W k _ l a k _ 1 -1- C k a k h- W k + l a k + 1 + • • • q- w n a n -1- ( d k -
=
Ck ) a k
w~ + • • • + Wk-~ + Ck + Wk+~ + " " " + W. + (dk -- Ck)
(2)
That is, the value of fL(Wj, W2, • • • , W. I d~) can be expressed as a fraction of which the numerator and denominator are the sum of numerators and denominators of w l a I -t- • • • q- W k _ l a k _ W 1+"
. .+Wk_
1 q-
Ckak + Wk+~ak+~ + " " •
l+c
kWwk+l+'"
which is the value of fe(w~, w2. . . . .
.+W
+ Wnan
n
wn I ck), and
(dk - Ck)ak dk -- Ck
respectively. When the value of wk changes from ck to dk, the value of fL changes from W, I Ck) to the right side of (2). Before continuing further exploration, a lemma is given. Since the proof is simply an arithmetic operation, it is omitted.
fL(W~, WZ. . . . .
Lemma 1. T h e f o l l o w i n g relations h o l d , p r o v i d e d that a > 0 a n d c > O. (a) I f b /a > d / c , then (b + d ) / ( a + c) < b /a, a n d i f b /a < d / c , then (b + d ) / ( a + c) > b /a. (b) F u r t h e r s u p p o s e a > c a n d b > d. T h e n b / a > d / c w h i c h i m p l i e s (b - d ) / ( a - c) > b / a ,
and
b /a < d / c w h i c h i m p l i e s (b - d ) / ( a - c) < b /a.
In f L ( W l , W2 . . . . . equivalent to (dk
--
dk-
Wn), we have ak = (d~ - c k ) a k / d k -- Ck. If ak
ck)ak< wlal
d- • • • "Jr" W k _ l a k _ 1 "4- C k a k -~ W k + l a k + 1 -t- • • • -~- W n a n
ck
wt + • • • + Wk_l + ck + wk+j + • • • + w ,
By Lemma l(a), we have wla 1 + • • • + Wk_lak_ W 1 +
1+
cgag +
Wk+lak+
1 + • • • + w.a.
" " " Jr- W k _ 1 " ~ C k " ~ W k + 1 "]- " " " " ~ W n
< wla I + • • • + Wk_lak_
1 d- C k a k + W k + l a k +
"q-
+ (d k -
(dk - -
Ck)ak
Ck )
1 + • • • q- w n a n
W l .~_ . . . ~t- W k _ l ..[- C k ..~ W k + l -]_ , o , .~_ W n
that is fL(W~, W2, . . . , W, I dk)
wn I Ck) then
w~ I Ck), a n d i f
From Theorem 3, by starting from the evaluation of fL(C~, C2 Cn) , we can find the minimum of fL. It will diminish the value of fL by replacing Ck with dk, then ak is less than the present value of fLSimilarly, in finding the maximum o f f u , we can start with evaluating f u ( c t , c2 . . . . . c,); if bk is greater .
.
.
.
.
Tian-Shy Liou, Mao-Jiun J. Wang / Fuzzy weighted average
310
than the present value of fu, then replace Ck with dk and this will increase the fu value. It is impossible to replace dk with Ck again when Ck has been replaced by dk. This is summarized in Theorem 4. Theorem 4. In the processes of finding the minimum of fL and the maximum of fv, Ck can never replace
dk again once it has been replaced by dk. Now, the last question needed to be answered in the IFWA algorithm is which candidate should be selected in replacing Ck by dk when more than one candidate is found. Let RL and Ru be the sets of the index i for which w~= di in fL(Wl, W2. . . . , Wn) and fu(w~, w2. . . . . w~) respectively. That is, wi = di for i e RL (Rtj) and wi = ci for i ~ RL (Rv). The selection rules are described in Theorem 5. T h e o r e m 5. In the process of finding the minimum of fL, suppose that L =fL(W~, W2. . . . .
Wn), where wi = di for i ~ RL and wi = ci for i ~ RL, is the present value of fL. Let I = {i I ai < L, i ~ RL} and fL( wl' W2 . . . . .
Wn I din)
-- m i n f L ( w l ,
i~1
w2. . . . .
wn
I di),
then dm is to be selected to replace C m . Similarly, in the process of finding the maximum of fv, suppose that U =fu(wl, wz, • . •, wn), where wi = di for i ~ R u and wi = ci for i ~ Ru, is the present value of fu. Let J = {j I bi > U, i ~ R v } and f u ( wl' W2 . . . . .
Wn I din) = m a x f u ( w l , w2 . . . . jeJ
, wn l d J) ,
then d,, is to be selected to replace c,,. Theorem 5 provides not only the selection rules when more than one index is involved in I or J, but it also provides the stopping rule for the IFWA algorithm when both I and J are empty. 2.2. IFWA algorithm The IFWA algorithm is developed by modifying steps 3 and 4 of the FWA algorithm. The algorithm is described in the following:
Step a. Evaluate L = f L ( C l , C2, . . • , C n ) and U = f u ( C l , C 2 . . . . . C n ) . Let I = {i I ai < L, i = 1, 2 . . . . . n}, J={j[bj>U,j =l,2,...,n} andRL=RU=O. Step b. If I = 0 then L is the minimum of fL and stop step b. Else w, = di for i ~ RL and wi = c~ for i ~t RL. bl. Evaluate li =fL(Wl, W2. . . . . W, t d~) for i ~ I. b2. L =lm = min/El li, put m into RL and take m away from I; put T = {i [ ai > L, i ~ I}. b3. Take i away from I, where i belongs to T. If 14: 0, then repeat step b, else stop step b and L is the minimum of fL. Step c. If J = 0 then U is the maximum of fu and stop step c. Else w~= d i for i ~ Ru and w/= c~ for i¢Ru. cl. Evaluate uj=ftj(wl, w2 . . . . . wn ]d r) for j eJ. c2. U =Um = maxj~j Uj, put m into Ru and take m away from J; put T = {j I b~ < U, j e J}. c3. Take j away from J, where j belongs to T. If J #=0, then repeat step c, else stop step c and U is the maximum of fv. Step d. [L, U] is the interval of a~i-cut of y. Go to step 5 of the FWA algorithm.
Tian-Shy Liou, Mao-Jiun J. Wang / Fuzzy weighted average
311
3. Example 3.1. Three-term weighted average Consider the weighted average
y = f ( x l , x2, x3, Wl, w2, w 3 ) -
WlX 1 + W2X 2 + W3X 3
wl + w2 + w3
where the fuzzy number A~ in x~, A2 in x2, A3 in x3, WI in w~, W2 in w2, and W3 in w3 are defined as {21,
UAI(X1)=
X1' ~ x 3 - 4,
ua3(x3) = [ 6 - x3 '
0<~Xl< 1, 1 ~
{x2-2, U'42(X2)=I.4--X2 '
4~
Wl/0.3, Uw,(Wl) = ( 0 . 9 - wO/0.6,
0 ~< wl < 0.3, 0.3 ~< wl ~< 0.9,
= ~ ( w 3 - 0.6)/0.2,
0.6~< w3 < 0.8, 0.8 ~< w3 ~< 1.
{(w2-0.4)/0.3, Uw2(Wz)={(1-w2)/0.3,
0.4 ~< w2 < 0.7, 0.7 ~< wz<~ 1,
2~
Uw3(W3) [(1_w3)/0.2,
In the F W A algorthm, it needs 22×3 = 64 evaluations on f for each a~j e [0, 1] to get the c~j-cut of y. For the new I F W A algorithm, with r e = 0 . 5 , the procedure and results are illustrated in the following section. The procedure is the same for all other tr/s.
3.2. Numerical result For tr = 0.5, the intervals of x~, x2, x3, w~, w2 and w3 are [a~ = 0.5, bl = 1.5], [a2 = 2.5, b2 = 3.5], [a 3 = 4.5, b3 = 5.5], [Cl = 0.15, d l = 0.6], [c2 = 0.55, d2 = 0.85], and [c3 = 0.7, d3 = 0.9] respectively. fL(Wl, W2, W3)=
0.5wl + 2.5w2 + 4.5w 3 W 1 + W2 +
and
fu(w~, w2, w3)=
1.5W1+3.5~+5.5%
W3
wl + WE+ W3
The computational procedure is as follows:
Step a: L =ft.(Cl, c2, c3) = fL(0.15, 0.55, 0 . 7 ) =
U = f u ( c l , c2, c3) = fu(0.15, 0.55, 0.7) 1={1,2},
J={3}
and
0.15X0.5+0.55X2.5+0.7X4.5 0.15+0.55+0.7
m
4.6 1.4'
0.15 x 1.5 + 0.55 x 3.5 + 0.7 x 5.5
6
0.15 + 0.55 + 0.7
1.4'
RL=Ru=0.
Step b: I = {1, 2} 4: 0, L = 4.6/1.4, Re = 0, so wl = cl, w2 = c2, w3 = c3. Step b l : l, = M 0 . 1 5 , 0.55, 0.7 I d , ) = fL(0.6, 0.55, 0.7)
4.6+(dl-cl)al 1.4 + (d~ - c1)
4.6+(0.6-0.15)
x0.5
1.4+(0.6-0.15)
4.825 1.85 '
12=•(0.15, 0.55, 0.7 1dz ) =fL(0.15, 0.85, 0.7) 4.6+(de-cz)a2_4.6+(0.85-0.55)×2.5 1.4 + (de - c2) 1.4 + (0.85 - 0.55)
5.35 1.7
Step b2: L = min{4.825/1.85, 5.35/1.7} = 4.825/1.85 = 11, i.e. m = 1, so R L = { 1 } , 1 = { 2 } . T=0.
Step b3: I = {2}, repeat step b.
And
Tian-Shy Liou, Mao-Jiun J. Wang / Fuzzy weighted average
312
Step b: I = {2} 4:0, L = 4.825/1.85, RL = {1}, so Wl = d l , w2 = c2, W3:C3. Step b l : 12 = fL(0.6, 0.55, 0.7 I d2) = fL(0.6, 0.85, 0.7)
Step Step Step Step
4.825 + ( d 2 - c2)a2
4.865 + (0.85 - 0.55) x 2.5
5.575
1.85 + (d2 - c2)
1.85 + (0.85 - 0.55)
2.15
b2: L = min{5.575/2.15} = 5.575/2.15 = 12, i.e. m = 2, so RL = {1, 2}, I = 0. And T = 0. b3: I = O, stop step b and L = 5.575/2.15 = 2.59 is the minimum offL. c: J = {3} 4:0, U = 6/1.4, R u = 0 , so W I = C 1 , W2~--C2, W 3 : C 3. c1:
u3 =fu(0.15, 0.55, 0.7 [ d 3 ) = f u ( 0 . 1 5 , 0.55, 0.9)
6+(d3-c3)b 3 6+(0.9-0.7) 1.4 + (d 3 - c3)
x5.5
1.4 + (0.9 - 0.7)
7.1 1.6"
Step c2: U = max {7.1/1.6} = 7.1/1.6 = u3, i.e. m = 3, so Ru = {3}, J = 0. And T = 0. Step c3: J = 0, stop step c and U = 7.1/1.6 = 4.44 is the maximum o f f u . Step d: [2.59, 4.44] is the interval of the 0.5-cut of y. Go to step 5 of the F W A algorithm. The procedures are repeated for tr = 0 and 0~= 1, and the results obtained are [1.68, 5.43] and [3.56, 3.56] respectively. These results are exactly the same as that of the F W A algorithm (see Dong and Wong [2]).
4. Discussion and conclusion
4. I. Advantage of the I F W A algorithm in calculation In the I F W A algorithm, the worst case is that all the left end-points ai are less than the initial L and all the right end-points bj are greater than the initial U, and no i in T (i.e. T = 0) is away from I in step b3, and no j in T (i.e. T = 0) is away from J in step c3, when steps b and c are repeated. In the worst case, the number of evaluations for fL and fv is 1 + ½n(n + 1) respectively. The total number if n(n + 1) + 2 which is much less than the number required for the F W A algorithm (22n). The former is polynomial time but the latter is exponential time. In evaluating the number of arithmetic operations involved in f, the F W A algorithm needs n multiplications, 2(n - 1) additions, and 1 division. In the I F W A algorithm, no matter how large the n is, there are only 1 multiplication, 1 subtraction, 2 additions, and 1 division. In here, we preserve the numerator and denominator of L, add them to (di-ci)ai and d i - c ~ in step bl to evaluate fL(Wl, W2. . . . . Wn I d~), and preserve the numerator and denominator of U, add them to (di - c~)a~ and di - ci in step cl to evaluate fu(wl, w2. . . . . wn I di) (see (2) and Section 3.2). As can be seen, the number of arithmetic operations has been reduced greatly.
4.2. Advantage of the IFWA algorithms in comparison For the number of comparisons, the I F W A algorithm needs at most n(n + 1) comparisons to find I and J in step a and to find T in steps b2 and c2. Also in steps b2 and c2, the maximum number of comparisons needed to find L and U is n - k (see Horowitz and Sahni [3], the straightforward maximum and minimum method) in the k-th cycle respectively, which has a total of n ( n - 1) comparisons. Overall, the algorithm requires at most 2n 2 comparisons to find the interval of the ag-cut of y. In the F W A algorithm, the maximum and minimum of y~, Y2, - . . , Yz2- are to be found. In analyzing the complexity of an algorithm, the number of element comparisons is always the main concern (see Borodin and Munro [1] and Horowitz and Sahni [3]). In finding the maximum and minimum value,
Tian-Shy Liou, Mao-Jiun J. Wang / Fuzzy weighted average
313
using the divide-and-conquer strategy, when the n u m b e r of elements m is a power of 2, i.e. m = 2 k for some positive integer k, the n u m b e r of comparisons is 3m - 2 . A n d it has been proven that no algorithm involves less than 3m - 2 comparisons (see Horowitz and Sahni [3], and Pohl [4]). In this case, m = 22". So the n u m b e r of element comparisons in the F W A algorithms is no less than 3 ( 2 2 " ) - 2 , that is far greater than 2n 2 comparisons involved in the I F W A algorithm. In summary, the I F W A algorithm which only modifies steps 3 and 4 of the F W A algorithm has been proven to be a more efficient algorithm in terms of the n u m b e r of calculations and comparisons required than that of the F W A algorithm.
Appendix Proof of Theorem 1. Since ai <<-x i <~ b i and w i > O, for i = 1, 2 , . . . , n, we have w i a i ~ w i x i <~ w~b~, for i = 1, 2 . . . . . n. T a k e the summation on the inequality, then divide by wl + w2 + • • • + wn and the proof is completed. Proof of Theorem 2. Since m i n f ( x l , x2, • • • , xn,
wl, w2, • • • , wn) <~f(xl,
x2 ....
, x,,
wl, w2, . • • , w~)
for all , Xn, Wl, WZ . . . .
( X l , X2 . . . .
e{(Xl,X2
.....
x.,wl,
, Wn) w2 . . . . .
we have m i n f ( x l , x 2 . . . . . x,,, wl, i = 1, 2 . . . . . n. This implies that
Wn) l a i < ~ x i < ~ b i ,
w2 . . . . .
m i n f ( x l , x2, . . • , x., wl, w2 . . . . . that is minf(x~,
x2 .....
minfL(w~, w2 . . . . .
w,,) <~f(al,
a2 . . . . .
w , , ) <<-minfL(Wl,
w.) ~< m i n f ( x l , x2 . . . . .
,n},
a., w~, wz .....
w,,)
a., wl, w2 . . . . .
w.),
w.) ~< m i n f ( a l , a2 . . . . .
x,,, wl, wz .....
i=1,2 ....
ci<~wi<~di,
w2 . . . . .
x., w~, w 2 , . . . ,
for all
ci <~ w~ <~ d i ,
w.). Following T h e o r e m 1, we have w.),
and we obtain that m i n f ( x l , x2 . . . . . x., wl, w 2 , . . . , w.) = minfL(wl, w2 . . . . . w.). Similarly, m a x f ( x ~ , xz . . . . . x., w~, w2 . . . . . w.) = m a x f v ( w l , w2 . . . . . w.) can be obtained. Thus T h e o r e m 2 is proven.
Proof of Theorem 4. Without losing generality, assume that R L = { 1 , 2 , . . . , k} and the value of
fL
fc(dl .....
is
fL(dl ..... dk-1,
d k , Ck+l . . . . .
Ck . . . . .
C,,),
d~al + • • • + dk-lak-~
C,)
when
cg
has
been
replaced
by
d~.
Because
ak<
SO
+ ckak + " " " +Cnan a k > O,
d~ + " • • + d k - ~ + C~ + • • • + Cn
which implies that dl(a
1-
ak) +...
4- d k _ l ( a k _ 1 -- a k ) q- Ck+l(ak+ 1 -- a k ) + " • • + Cn(an -- a k ) > O, d~ + .
• • +dk-~
+ck
+" • • +c,
and thus d l ( a l - ak) + ' "
Let RL = {1, 2 , . . .
+ dk_l(a~_~ - ak) + Ck+~(ak+~ -- a~) + . . .
+ c . ( a . -- a~) > 0.
, p } , where p ~
fL(dl .....
(A1) d p , Cp+l . . . . .
cn).
Tian-Shy Liou, Mao-Jiun J. Wang / Fuzzy weighted average
314
Surely, p/> k, and fL(dl . . . . . dl(al
d i a l + " • • + dkak + " " " + dpap + cp+lap+l + • • • + c . a ~
tip, Cp+l, . . . , % ) - - a k = ak) + ' ' "
-
+ dk-l(ak-1
dl + " ' " + d p +Cp+l + ' " • + c.
--ak
-- a k ) -I- d k + l ( a k + 1 -- a k ) + . . .
=
+ d p ( a v - ak) + Cp+l(ap+l - ak) + ' ' '
+ cn(an -- ak)
d l + " • " + dp + Cv+l+" • • +c,, >I d l ( a , - ak) + ' ' "
+ dk-l(ak-1
-- ak) + Ct,+l(ak+l -- ak) + ' ' "
+ c,,(a,, - ak) > 0
d l + " " • + dp + cp+l + " " " + c,,
(see (A1), and d i >! ci, i = k + 1 . . . . . is less than the present value of fL. f
(d, . . . .
,
d k - a , ck, dk+~ . . . . .
dial + • • • + dk-lak-i
p ) . It is proven that ak
dp, cp+l . . . . .
c,,), that is ak
c.)
dp, Cp+l . . . . .
+ Ckak + dk+~ak+l + " " " + dpap + Cp+lap+ 1 + • • • + Cna n
d l + " • " + d k - I + ck + d k + l + " " " + dp + Cp+ 1 "q- " " " -~- Cn
= dial + • • • + dpa. + c.+la.+l dl + " " + df, + Ce+l + >
d l a 1 + . • • + d p a , + Cp+lap+ 1 dl +...+
=fL(dl,
• • • ,
+ " • • + c . a ~ - (dk - c k ) a k •
•
•
+ C. -- (dk -- Ck)
+" • • + %a.
(from Lemma l(b))
dp q-Cp+l + . • . + C n
dk-1, dk, dk+l,..
•,
dp, Cp+l . . . . .
Cn).
This proves that the replacement of dk by Ck would increase the present value of rE- Therefore, in finding the minimum of rE, Ck can never replace d k again once it has been replaced by d k. Similarly, the same reason is true for finding the maximum of fu. The proof of Theorem 4 is completed•
Proof of Theorem 5. To simplify the problem, we suppose that only as and aj are less than the present value of fL(W~, W2 . . . . . W , ) , where wk = d k for k e R~ and wk =Ck for k ~RL. It is obvious that i,j~RL. Then di may replace ci to get fc(Wl, W Z , . . . , w , l d i ) or dj may replace cj to get fL(Wl, WE,...,
W, ] dj).
Assume that f L ( W , , WE. . . . .
w , l di)
(A2)
w~ l di).
Case 1: If we select dj to replace cj, and suppose ai
W~ ] dj), then d~ replaces ci to get
f c ( W l , W2, . . . , Wn I dj, di), and fL(Wl,
WE,...,
Wn I dy, di) ~ f L ( W , , W2, . . . , Wn I d j ) .
Two consequences may occur: (1) f L ( W , , W2, . • • , W , ] dj, di) <~fL(W~, W2, • • • , W, d~); this is what we want• (2) f~(Wl, WE, • • •, W~ I dj, d~) >fL(w~, W2. . . . . Wn d~); this is worse than replacing c~ by di only and it is not a desirable outcome. Case 2: If we select d; to replace c~, the result can be one of two outcomes: (1) aj ~>fL(Wl, W2. . . . . W, [ di); by Theorem 3, dj does not replace cj. (2) aj < fL(W~, W2, , W, ] di); in this c a s e , dj r e p l a c e s cj to g e t f L ( w , , w2 . . . . , wn ] d~, dj) a n d f L ( W I ' W2 . . . . .
Wn I d i ' d j ) < - f L ( W , , W2 . . . . .
By (A2), we have f L ( W l , W2 . . . . .
w , I di)•
W~ ] d~, dj) <~fL(W~, WE . . . . .
W~ [ dj), which is just what we want.
Tian-Shy Liou, Mao-Jiun J. Wang / Fuzzy weighted average
So, in finding fL(Wl, W2. . . . . W. M w ~ , w2 . . . . w. This relationship proven.
the minimum of dj). Similarly, in d 3
315
fL, di is to replace ci when f L ( W l , W2 . . . . , W. 14)< finding the maximum of fu, dj is to replace c~ when w. l dj). the case of more than two candidates, and Theorem 5 is thus
References [1] A. Borodin and I. Munro, The Computational Complexity of Algebraic and Numeric Problems (American Elsevier, New York, 1975). [21 W.M. Dong and F.S. Wong, Fuzzy weighted averages and implementation of the extension principle, Fuzzy Sets and Systems 21 (1987) 183-199. [3] E. Horowitz and S. Sahni, Fundmentals of Computer Algorithms (Computer Science Press, Rockville, MD, 1978) 108-112. [4] I. Pohl, A sorting problem and its complexity, Comm. Assoc. Comput. Mach. 15 (1972) 462-463. [5] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning, Inform. Sci. 8 0975) 199-249.